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Security Analysis

Using the properties of dynamical systems, we can evaluate cryptologic security as follows [5,7,8].

Sensitive dependence on Initial Conditions

A sufficient condition that two ciphertexts from a pair of adjacent plaintexts (or keys) be independent is

$\begin{displaymath}n = 2.39 \log_2 M + 15. \end{displaymath}$

If A and M are relatively prime, we can reduce n to

$\begin{displaymath}n=1.66\log_2 M+15. \end{displaymath}$

Exponential Information Decay

We can analytically calculate the correlation between plaintexts and ciphertexts using the invariant measure and the Perron-Frobenius operator [3,4].

Assuming uniform distribution of plaintexts, we can calculate the correlation function between plaintexts and ciphertexts. The correlation decay is exponential: (2a-1)ⁿ. The decay rate (2a-1)ⁿ may be universal to be relevant in various properties of the encryption function F_A such as the invariant measure, the quantity to represent the strength of the mixing property, and the KS entropy.

Bitwise Independence

The probability that the i-th bit of a plaintext is 1(or 0) and the j-th bit of a ciphertext is 1 (or 0) is $\frac{1}{4}$ in the limit as $n\to\infty$ . In other words, a bit of a plaintext and one of a ciphertext are independent of each other. To derive the above result, we benefit from the fact that f_a is mixing [4].

Evaluation of Iteration Number by the KS Entropy

We can apply the KS entropy h_KS [1,3]

$\begin{displaymath}h_{KS}(\tilde{f_a}^n) = n\lambda \cong H(P) = \log \vert P\vert \end{displaymath}$

to evaluate the required iteration number n. The KS entropy is in conjunction with the information theoretical entropy.

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